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Local Maxima of the Potential Energy on Spheres

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Let S d be a unit sphere in ℝd+1, and let α be a positive real number. For pairwise different points x 1,x 2, . . . ,x N S d, we consider a functional E α (x 1,x 2, . . . ,x N ) = Σ ij ||x i x j ||α. The following theorem is proved: for αd − 2, the functional E α (x 1,x 2, . . . ,x N ) does not have local maxima.

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Correspondence to D. V. Radchenko.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 10, pp. 1427–1429, October, 2013.

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Radchenko, D.V. Local Maxima of the Potential Energy on Spheres. Ukr Math J 65, 1585–1587 (2014). https://doi.org/10.1007/s11253-014-0880-4

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  • Potential Energy
  • Local Maximum
  • Point Theorem
  • Unit Sphere
  • Positive Real Number